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We consider the problem of keeping an arbitrary state $\rho_s$ out of thermalequilibrium. We find that counter-acting thermalisation using only a resourcesystem which is in a stationary state at the initial time and a system-resourceinteraction that preserves the global energy is possible if and only if thetarget state $\rho_s$ is block-diagonal in the eigenbasis of the system'sHamiltonian $H_s.$ As a consequence, we compute the extra work the operatormust provide by tuning the resource-system interaction to overcome thisconstraint. This quantity, which is interpreted as the work needed to preservethe coherences in the state, can be expressed in terms of the target state$\rho_s$ and the thermal equilibrium state $\rho_{\beta},$ and it isproportional to the symmetrized relative entropy between $\rho_s$ and$\rho_{\beta}.$

The work cost of keeping states with coherences out of thermal equilibrium